Math Problem Statement
Through elementary operations, prove that the rank of above matrice is 2
Solution
To prove that the rank of the matrix is 2 using elementary row operations, we will convert the matrix into its row echelon form (REF) and then count the number of non-zero rows. The number of non-zero rows in REF gives the rank of the matrix.
Starting with the matrix:
2 & 4 & 8 & 6 \\ 1 & 2 & 1 & 0 \\ 3 & 6 & 6 & 3 \end{bmatrix}$$ Let's apply the elementary row operations step by step.from sympy import Matrix # Convert the matrix to a sympy Matrix to perform elementary row operations matrix_sympy = Matrix([[2, 4, 8, 6], [1, 2, 1, 0], [3, 6, 6, 3]]) # Perform row reduction to get the row echelon form (REF) ref_matrix = matrix_sympy.echelon_form() ref_matrixAfter performing elementary row operations, the matrix in row echelon form (REF) is: $$\begin{bmatrix} 2 & 4 & 8 & 6 \\ 0 & 0 & -6 & -6 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ In this form, we observe that there are **2 non-zero rows**. Therefore, the rank of the matrix is $$2$$, as previously determined. Would you like to explore more about the steps or the theory behind the row reduction process? Here are some related questions: 1. What elementary row operations were used to reduce the matrix? 2. Why do zero rows not contribute to the rank of a matrix? 3. How can the rank of a matrix help determine whether a system of equations has a unique solution? 4. What is the difference between rank and nullity in a matrix? 5. How does the echelon form relate to solving linear systems? **Tip:** When reducing a matrix to row echelon form, pivot elements (the leading non-zero entries in each row) play a key role in determining the rank.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Row Echelon Form
Rank of a Matrix
Formulas
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Theorems
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Suitable Grade Level
Advanced High School